Theoretical study of symmetric and antysymmetric plasmons in chains of coupled plasma cylinders

Theoretical study of plasmon resonant eigenfrequencies in coupled plasma cylinders is presented. Mechanism of the plasmonic mode coupling that can be considered as symmetric and antisymmetric combinations of isolated cylinders plasmons is investigated. Accurate analysis of the spectrum of different plasmon resonances is presented.


INTRODUCTION
Metallic nanostructures are the subject of growing interest in recent years due to the possibility of a strong light localization beyond the diffraction limit via the excitation of surface and local plasmons [1][2]. Various elements such as plasmonic waveguides [3][4], subwavelength resonators [5][6] and optical nanoantennas [7][8][9] have been studied recently. Surface and localized plasmons have been explored for their potential in single molecule detection including use of the Surface Enhanced Raman Scattering (SERS) effect [10][11], transmissions through the subwavelength apertures [12][13], subwavelength imaging [14] etc. Plasmonic structures of different shapes (nanowires, nanorods, nanospheres, nanoshells) can be produced by various fabrication techniques. The silver nanowire structure is a candidate for key components in future ultra-compact photonic devises [15]. It can be considered as a plasmon biosensor to monitor tiny biomolecular interactions [16] and as a novel modulator to control the intensity of the transmitted surface plasmon polaritons through a nanowire array [17]. Possible future nanophotonic technologies demand devices that can generate stimulated emission through the excitation of the surface plasmons [18][19]. For these applications an accurate frequency domain modeling that provides a valuable insight into fundamental processes is of great importance.

II. PROBLEM FORMULATION AND METHOD OF SOLUTION
In this paper we solve the eigenvalues problem for a chain of coupled plasma cylinders (columns). Radius of each column is a , separation distance between them is d and plasma is described by the permittivity p ε that is given by the Drude model: 2 1 ( ) 1 ( ( )) p p i ε ω ω ω ω γ − = − ⋅ + (1) Here p ω represents the plasma frequency, γ is the material absorption. Sub-wavelength resonances are possible when  Unknown coefficients s K and s M are found from the boundary conditions, requiring the continuity of the tangential components of the total electric and magnetic fields at each cylindrical column's surface. Using the addition theorem for the Bessel functions we arrive to an infinite system of algebraic equations that can be truncated in order to provide a controlled numerical precision. We have to mention that all eigenfrequencies are complex i , where 0 ω′′ < represents damping and ω′ is associated with the eigen oscillation frequencies. Quality (Q) factor of plasmons can be evaluated through the formula 2

III. RESULTS
Eigenvalue problem in an isolated plasma cylinder is treated analytically. Figure 1 illustrates the value of the real part of plasmon eigenfrequency versus normalized frequency    For the case of two coupled plasma cylinders the structure has two symmetry axes that causes four classes of excited plasmons with different symmetry: EE (even symmetry with respect to x and y axes), EO (x -even; y -odd), OE (x -odd; y -even), OO (x -odd; y -odd). Consequently the plasmonic modes of the coupled plasma cylinders can be viewed as symmetric and antisymmetric combinations of plasmons of isolated cylinder.  The Q-factor of dipole plasmons in two coupled plasma cylinders ( 1 s = ) is shown if Fig. 4. We examined the range of normalized separation distance from 0 to 30. For distant plasma cylinders Q of coupled plasmonic modes is evidently smaller than Q factor of corresponding plasmons of the isolated cylinder. Peaks of Q are observable for the case when separation distance tends to the wavelength. Adding one more plasma cylinder does not alter the number of the symmetry axes of the structure however bonding and antibonding combinations of the plasmons can be excited. Figure 5 demonstrates the near-field distributions of different plasmons of three coupled plasma cylinders for different number of angular field variations and for different values of the normalized separation distance between the coupled cylinders. 1 s = (dipole modes) and 2 s = (quadrupole modes), respectively. We see the appearance of bonding and antibonding modes in EO and OE symmetry classes for dipole modes ( 1 s = ). However for quadrupole modes ( 2 s = ) we observe reverse situation that means appearance of bonding and antibonding modes in EE and OO classes. It can be seen that the upward shift in frequency is much faster than downward shift for both dipole and quadrupole modes (see Figs. 6 and 7). We compare the Q-factors of dipole and quadrupole plasmons in chain of three plasma cylinders in Figs. 8 and 9. The growth of Q-factors for certain separation distances is observable. Dramatic enhancement in Q for quadrupole modes is evident. Figure 6. The normalized frequency versus the normalized separation distance between the three coupled plasma cylinders for different plasmons (s=1) and for isolated cylinder (s=1).
In a linear chain of four coupled cylinders bonding and antibonding modes appear in each symmetry class. We have to mention the total number of excided plasmons for each value of s twice greater that number of cylinders in a chain. Figure 7. The normalized frequency versus the normalized separation distance between the three coupled plasma cylinders for different plasmons (s=2) and for isolated cylinder (s=2).    Figure 10 presents eigenfrequencies of coupled EE dipole plasmonic modes in a chain of plasma cylinders. It is seen that for large separation distances between cylinders all modes converge to plasmon modes of an isolated plasma cylinder. With decreasing of the separation distance we observe upshifting of the frequency for bonding modes and down-shifting for antibonding modes. Figure 10. The normalized frequency versus the normalized separation distance between the coupled plasma cylinders in a chain on N cylinders for EE plasmon (s=1) and for isolated cylinder (s=1). Figure 11. Q-factor for chain of N coupled plasma cylinders (s=1, p w =1) for EE plasmons and for isolated cylinder (s=1). Figure 11 shows the Q factor of a variety of EE plasmons. We see the growing of Q with increasing the number of cylinders in a chain, besides the Q for antibonding plasmons exceed those for bonding ones.

IV. CONCLUSIONS
The eigenfrequencies of the coupled cylinders filled with negative permittivity plasma have been analyzed. It has been shown that individual plasmons of isolated column interact and form symmetric and antisymmetric plasmonic coupled modes of different types. Accurate analysis of the influence of the coupling of plasma cylinders on their spectrum of different plasmon resonances is presented.